Let \(O\subset {\mathcal {M}}\)  be the set of points where the gradients of \(\hat{\phi }\)  and \(\phi _{\epsilon }\)  (for the entire countable sequence \({\epsilon }\) ) are not defined, then \(O\)  is of volume-measure zero on \({\mathcal {M}}\) . Indeed, the functions \(\hat{\phi }, \phi _{\epsilon }\)  are differentiable almost everywhere on \({\mathcal {M}}\)  by Lemmas 2 and 4 in [1]}.
Furthermore, if we denote by \(\hat{t}\)  the optimal transport defined by \(\hat{\phi }\) , as discussed in Chapter 13 in [2]} the set of all \(x\in {\mathcal {M}}\)  for which \(\hat{t}(x)\)  belongs to the cut locus is of measure zero. We add this set to \(O\) , keeping it of measure zero.
